Consider the sentence  (∃xP(x)∧∃xQ(x))→∃x(P(x)∧Q(x))
 . Which of the following is true?

Question 4Answer

A.
The sentence is valid.


B.
The sentence is not valid. It is false under the interpretation where  x
  ranges over the positive integers,  P(x)
  is " 7
  divides  x
 " and  Q(x)
  is " 11
  divides  x
 ".


C.
The sentence is not valid. It is false under the interpretation where  x
  ranges over the positive integers,  P(x)
  is " 8
  divides  x
 " and  Q(x)
  is " x
  is odd".


D.
The sentence is not valid. It is false under the interpretation where  x
  ranges over the positive integers,  P(x)
  is " 8
  divides  x
 " and  Q(x)
  is " 4x
  is odd".

The given logical sentence is \((\exists x \ P(x) \wedge \exists x \ Q(x)) \rightarrow \exists x \ (P(x) \wedge Q(x))\).

To determine whether this sentence is valid or not, we need to evaluate the interpretations provided and see if they can make the sentence false.

### Interpretation of the Sentence:

- **Left-hand side (\(LHS\))**: \(\exists x \ P(x) \wedge \exists x \ Q(x)\)
  - This means there exists some \(x\) such that \(P(x)\) is true, and there exists some \(x\) such that \(Q(x)\) is true. Note that the \(x\) in \(P(x)\) and the \(x\) in \(Q(x)\) may be different.
  
- **Right-hand side (\(RHS\))**: \(\exists x \ (P(x) \wedge Q(x))\)
  - This means there exists some \(x\) such that both \(P(x)\) and \(Q(x)\) are true for the same \(x\).

The sentence \((LHS) \rightarrow (RHS)\) asserts that if \(LHS\) is true, then \(RHS\) must also be true. This is a statement about the existence of an \(x\) that satisfies both properties simultaneously, given that there exist \(x\) values that satisfy each property individually.

### Evaluating the Options:

**Option B**:
- \(P(x)\) is "7 divides \(x\)", meaning \(x\) is a multiple of 7.
- \(Q(x)\) is "11 divides \(x\)", meaning \(x\) is a multiple of 11.

For the \(LHS\) to be true, there must exist some \(x\) that is a multiple of 7 (for \(P(x)\)) and some (potentially different) \(x\) that is a multiple of 11 (for \(Q(x)\)). 

For the \(RHS\) to be true, there must exist some \(x\) that is a multiple of both 7 and 11 simultaneously (i.e., a common multiple of 7 and 11). The smallest such \(x\) would be 77, which exists. Therefore, \(RHS\) would also be true, meaning the whole statement is true under this interpretation.

**Option C**:
- \(P(x)\) is "8 divides \(x\)", meaning \(x\) is a multiple of 8.
- \(Q(x)\) is " \(x\) is odd".

For the \(LHS\) to be true, there must exist some \(x\) that is a multiple of 8 and some (potentially different) \(x\) that is odd. 

However, for the \(RHS\) to be true, there must exist some \(x\) that is both a multiple of 8 and odd. But no such \(x\) exists because multiples of 8 are even, never odd. Therefore, \(RHS\) is false, and the whole statement is false under this interpretation.

**Option D**:
- \(P(x)\) is "8 divides \(x\)", meaning \(x\) is a multiple of 8.
- \(Q(x)\) is "4\(x\) is odd".

For \(LHS\) to be true, \(P(x)\) and \(Q(x)\) must each have some \(x\) that satisfies them. However, for \(Q(x)\) to be true, \(4x\) must be odd, but \(4x\) is always even, so \(Q(x)\) can never be true. If \(Q(x)\) is never true, the \(LHS\) itself is false, so the implication (\(LHS \rightarrow RHS\)) would actually be true (because a false premise implies anything). Therefore, this option doesn't make the sentence false.

### Conclusion:

**The correct answer is C**. The sentence is not valid because it is false under the interpretation where \(x\) ranges over the positive integers, \(P(x)\) is "8 divides \(x\)," and \(Q(x)\) is " \(x\) is odd."

---

Would you like any further details on the reasoning? Here are some related questions you might find interesting:

1. How does the distributive property of quantifiers affect logical sentences?
2. What are some examples of logically valid sentences involving quantifiers?
3. How can counterexamples be used to disprove the validity of a logical statement?
4. How does one construct the contrapositive of a logical implication?
5. What are the different interpretations of existential and universal quantifiers in logic?

**Tip:** When analyzing logical statements, it’s important to clearly understand how quantifiers interact with the predicates in the expression.

Explore the logical implication (∃xP(x) ∧ ∃xQ(x)) → ∃x(P(x) ∧ Q(x)) in mathematics. Understand the validity of the statement under different interpretations of logical quantifiers. This detailed analysis covers scenarios involving divisibility and oddness of integers.

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